Algorithms for closed-loop feedback baseddistributed adaptive beamforming in wireless sensor

networksStephan Sigg, Michael Beigl

Institute of operating systems and computer networks, TU BraunschweigMuhlenpfordtstrasse 23, 38106 Braunschweig, Germany

{sigg,beigl}@ibr.cs.tu-bs.de

Abstract—We study a closed-loop feedback based approachto distributed adaptive transmit beamforming in wireless sensornetworks. Three algorithms to achieve sufficient phase synchro-nisation of carrier signals are considered. In particular, we studythe impact of normal and uniform distributions for the phasealteration probability on the performance of the synchronisationprocess. Both distributions are studied for various probabilitydistributions and variances. Furthermore, a local random searchheuristic is studied for various sizes of the search neighbourhood.

I. INTRODUCTION

A central issue in wireless sensor networks is the forwardingof gathered information from the network to an externalreceiver. When the receiver node is mobile and requestsinformation in a spontaneous ad-hoc manner, wireless trans-mission is required. As power consumption is typically a majorconstraint in wireless sensor networks, the transmission rangefor this transmission is strictly limited. However, by properlysuperimposing transmit signal components of distinct nodesthe transmission range of a sensor network can be increased.

This approach was studied by various research groups. Thesolutions proposed cover random schemes as multi-hop [1],[2], [3], Data flooding [4], [5], [6], [7] and cluster based[8], [9], [10] techniques as well as collaborative beamforming[11] or cooperative/virtual MIMO for wireless sensor networks[12], [13], [14], [15].

While the random schemes do not consciously impact thedirection of the transmission beam, in virtual MIMO, singleantenna nodes cooperate to establish a directed beam froma multiple antenna wireless network [13], [12], [14]. Whilevirtual MIMO has capabilities to adjust to different frequenciesand is highly energy efficient [15], [16], it requires accuratetime synchronisation, complex transceiver circuitry and signalprocessing that might surcharge the power consumption andprocessing capabilities of simple sensor nodes.

A simpler approach is the one-bit feedback closed-loopsynchronisation considered in [17], [18], [19]. The authors de-scribe an iterative process in which the source nodes randomlyadapt the carrier phases of their base band transmit signal.This random process is guided by a one-bit feedback on the

synchronisation quality that is computed by the destinationnode.

In [20] this process was also implemented and verified ina network of three transmitters and one receiver at 60 GHz.It was shown that this process converges to an optimum withprobability 1 [19]. Furthermore, the runtime of the approachis studied in [19], [21]. An asymptotic upper bound ofO(n ·k · log n) on the expected optimisation time in a networkof n nodes with k distinct phase offsets at the transmittingnodes was proved in [22]. In this consideration carrier phasesare altered uniformly at random from the transmitting nodesduring several iterations of the synchronisation process. Thisis in contrast to various prior studies that considered a normaldistribution on the phase alteration process [17], [18]. For thenormal distribution the expected asymptotic synchronisationtime was not yet derived. In [19] the expected synchronisationtime of this process was expected to rise linearly in the numberof nodes. However, a central assertion of the proof was leftopen. We study the impact of normal and uniformly distributedphase alteration processes on the optimisation speed of dis-tributed adaptive beamforming in wireless sensor networks.

II. DISTRIBUTED ADAPTIVE BEAMFORMING IN WIRELESSSENSOR NETWORKS

In distributed adaptive beamforming in wireless sensor net-works an individual node i in a network of size n cooperateswith other nodes to reach a distant receiver by superimposingits individual carrier signal <

(m(t)ej(2π(f+fi)t+γi)

)with

carrier signals transmitted from other nodes. The approachconstitutes a closed-loop feedback technique that is attained inan iterative process. During each iteration, following a randomdistribution, each transmit node i adds random phase andfrequency offsets γ′

i, f′i to its carrier phase γi and frequency

fi. According to the random distribution it is possible thatall, some or no node adjusts the phase or frequency ofits carrier signal. The nodes then transmit to the destina-tion simultaneously. From this received superimposed sumsignal <

(RSSsumm(t)

∑ni=1 e

j(2π(f+fi)t+γi))

the receiverestimates the level of phase synchronisation achieved (e.g.by SNR or comparison to an expected signal). At the end

Fig. 1. A point in the search space (configuration of transmit nodes) spannedby the phase offsets of the carrier signals s1 and s2

of each iteration, the receiver broadcasts this estimation as achannel quality indicator to the network. Nodes then sustaintheir (adapted) carrier phases γi + γ′

i and frequencies fi + f ′i

when the feedback has improved to the feedback obtainedduring the last iteration or else reverse them.

These steps are repeated until a stop criteria is met. Possiblestop criteria are, for example, the maximum iteration count orsufficient quality of the estimated channel.

The exact implementation of the phase alteration processin each iteration impacts the performance of the optimisationapproach. Generally, two parameters are of importance. One isthe number of nodes altering their carrier phase and frequencyin each iteration and the other is the probability distributionapplied on the alteration of phases and frequencies of onecarrier signal.

When we illustrate the search space of the algorithm withrespect to these parameters we observe that they effectivelyimpact the neighbourhood size of the search approach (seefigure 1). When the probability to alter the carrier phase ofa single node is low, few carrier signals are altered in oneiteration. In the figure this might translate to the situationthat the phase offset of only one carrier signal is altered. Thenew search point then differs from the recent one in only onedimension. When a uniform distribution is implemented, allpoints along this dimension are equally probable while fora normal distribution the most probable points are near tothe recent search point as specified by the variance of theprobability distribution utilised to alter phase and frequencyoffsets. When, however, the mutation probability is increased,so that more often several carrier signals (in the figure, forexample, both carrier signals s1 and s2) are altered, the newsearch point is drawn from a region in the search space. In thecase of the uniform distribution the whole search space mightconstitute this region. Consequently, a smaller probability fornodes to alter their phase and frequency or a smaller variancefor the random process to alter these parameters increases the

Fig. 2. Base configuration of the simulations conducted.

probability to draw a search point that is near to the recentone.

We expect that a moderate size of the neighbourhood (i.e.probability to alter phase or frequency of one carrier signaland variance for the random process to alter these values) isbeneficial at the start of the synchronisation. The initial searchpoint or the initial synchronisation of carrier phases is typicallyfar away from the optimum. Therefore, many search pointshave a higher fitness value (are better synchronised) than theinitial one. Consequently, since the probability to improvethe fitness value is high, it might be beneficial to increasethe neighbourhood size so that also bigger improvements arepossible. Later in the optimisation process, however, when theoptimum is near, most search points reduce the fitness valueso that the probability to increase it is best for search pointsdrawn from a small neighbourhood. The results obtained insimulation studies in section III also indicate towards thecorrectness of this assumption.

III. SIMULATION RESULTS

We have implemented the scenario of distributed adaptivebeamforming in Matlab for a network of 100 transmit nodesthat are placed uniformly at random on a 30m× 30m squarearea. The receiver is located 30m above the centre of this area(cf. figure 2).

The transmission power of transmit nodes is set to 1 mWwith no antenna gain at the transmitter or receiver. Environ-mental noise (AGWN) is set to −103 dBm as proposed in [23]and the pathloss is calculated by the Friis free-space equationas

Prx = Ptx

(λ

2πd

)2

GtxGrx

Receiver and transmit nodes are stationary. The base configu-ration for all simulations is summarised in table I. Frequencyand phase stability are considered perfect. We study the impactof a uniform or normal distribution of the phase alteration pro-cess on the performance of distributed adaptive beamformingin wireless sensor networks.

We experience good synchronisation after about 3000 itera-tions (cf. figure 4). For every configuration of the networkthe simulation we conducted 10 identical simulation runs

TABLE IBASE CONFIGURATION OF THE SIMULATIONS CONDUCTED. PRX IS THE

THE RECEIVED SIGNAL POWER, d IS THE DISTANCE BETWEENTRANSMITTER AND RECEIVER AND λ IS THE WAVELENGTH OF THE

SIGNAL

Property ValueNode distribution area 30m× 30mLocation of the receiver (15m, 15m, 30m)Mobility stationary nodesBase band frequency fbase = 2.4 GHzTransmission power of nodes Ptx = 1 mWGain of the transmit antenna Gtx = 0 dBGain of the receive antenna Grx = 0 dBIterations per simulations 6000Identical simulation runs 10Random noise power −103 dBm

Pathloss calculation (Prx) Ptx“

λ2πd

”2GtxGrx

with 6000 iterations. One iteration consists of the networktransmitting for few signal periods, feedback computation atthe receiver, feedback transmission and feedback interpretationat the network. It is possible to perform these steps within fewsignal periods. The time consumed for a complete synchronisa-tion of 6000 iterations is therefore in the order of millisecondsfor a base band signal frequency of 2.4 GHz.

Simulation results show the median and the standard devi-ation. Signal quality at the receiver is measured by the RMSEbetween the received signal and an expected optimum signal.

RMSE =

√√√√ τ∑t=0

(∑ni=1 si + snoise − sopt)

2

n(1)

In this formula, τ was chosen to cover several signal pe-riods. The terms si = RSSi<

(m(t)ej(2π(f+fi)t+γi)

)and

sopt = nRSSsum<(m(t)ej(2πfoptt+γopt)

)represent the i-th

signal component of the received sum signal and the expectedoptimum signal, respectively. The optimum signal is calculatedas the perfectly aligned and properly phase shifted receivedsum signal from all transmit sources. For the optimum signal,noise is disregarded. Figure 3 depicts the optimum sum signal,the initial received sum signal and the signal of synchronisedcarriers after 6000 iterations when phase alterations are chosenaccording to a per node uniform distribution with mutationprobability 1

n .Figure 4 illustrates the phase offset of received signal com-

ponents for an exemplary simulation run. In this simulation,phase alterations are chosen according to a per node uniformdistribution and nodes alter their phase with probability 0.01.For n = 100 transmit nodes this translates to an averagenumber of 1 transmitting node in every iteration. We observethat after 6000 iterations about 95% of all carrier signals havea relative phase offset of +/- 0.1π. The median of all variancesof the phase offsets for simulation runs with this configurationis 0.2301π after 6000 iterations.

In the first set of simulations we considered the impact ofthe mutation probability on the synchronisation performancewhen new transmit phase offsets are drawn according to auniform distribution.

Fig. 3. Received sum signal from 100 transmit nodes without synchronisationand after 6000 iterations. Phase alterations are drawn uniformly at randomwith mutation probability 1

nfor each node

Fig. 4. Evolution of the phase adaptation process for uniform distributionof phase adaptation and 1 percent nodes change

When the average number of carrier signals that are mod-ified in one iteration is altered, this affects the performanceof the optimisation approach, as the stepwidth of the randomprocess is changed. We are interested in the optimum per-centage of nodes that alter their phase per iteration (mutationprobability) when the phase-alteration for each mutating nodeis drawn according to a uniform distribution. Figure III depictssimulation results for several mutation probabilities.

A mutation probability of κ represents the case that onaverage κ · n out of n transmit signal components mutate(randomly alter their phase offset) in one iteration.

We observe that with a low mutation probability the over-all performance of the synchronisation process is improved.At the start of the optimisation, however, higher mutationprobabilities are more beneficial. This accounts for the factthat initially many configurations exist that would improve thefitness value as the initial point is not well synchronised withhigh probability (cf. section II).

Since a mutation probability of 0.01 translates to 1 nodeto adapt the phase offset of its carrier signal per iteration on

Fig. 5. Performance of distributed adaptive beamforming in a wireless sensornetwork of 100 nodes and normal and uniform probability distributions on thephase mutation probability. Uniform distribution of phase mutations.

Fig. 6. Normal distribution of phase mutations with mutation probability 1

average, smaller mutation probabilities are not beneficial.We also considered a normal distribution for the phase

alteration as it was applied in [20], [24], [19], [25] andstudy the impact of the mutation probability and the varianceutilised.

Figure 6 shows the performance of this approach withseveral values for the mutation variance applied to the phasemutation process when all nodes mutate in every iteration(mutation probability of 1). This configuration is identical tothe simulations conducted in [20], [24], [19], [25]. For easeof presentation, error bars are omitted in this figure.

We observe that a smaller variance is beneficial for theperformance of the synchronisation process. The reason forthis is again the reduced neighbourhood size of the searchmethod at smaller variances. In particular, by means of themutation variance the neighbourhood size can be adapted toan optimum for a given mutation probability.

We approximated the optimum variance for several mutationprobabilities. Generally, a higher mutation probability necessi-tates a smaller variance to achieve the optimum performance.

Fig. 7. Performance of distributed adaptive beamforming in a wireless sensornetwork of 100 nodes and normal and uniform probability distributions on thephase mutation probability. Normal distribution of phase mutations.

Figure III depicts the performance for mutation probabilitiesof 0.2, 0.05 and 0.01 and near optimum variance, respectively.

For these configurations the performance is similar to theperformance of the uniform distribution with mutation proba-bility 0.01.

Summarising, we conclude that the best performance wasachieved by small mutation probabilities in which one transmitsignal component is altered in every iteration on average.This optimisation performance can be further improved at thestart of the synchronisation process when a moderate mutationprobability is implemented that is gradually adapted in thecourse of the synchronisation process.

For the normal distribution the performance is also impactedby the mutation variance so that for a fixed mutation probabil-ity the performance can be improved by adopting the varianceof the process.

Overall, results for near optimum configurations are simi-lar for normal and uniform distributed phase alteration pro-cesses. A straightforward implementation would, for instance,constitute an adaptive phase mutation process with uniformdistribution.

Since in [26] the search space of distributed adaptive beam-forming was classified as weak unimodal, we expect a nearestneighbour search approach to be best suited to achieve fastsynchronisation. We implemented and tested a local randomsearch heuristic for this scenario with various neighbourhoodsizes. The implementation differs from the normal distributionmainly since it utilises a fixed size neighbourhood instead of avariance on the phase alteration process. Figure 8 shows thatthe performance is similar to the implementation with uni-formly distributed phase alteration probabilities. We thereforeconclude that the global random search approaches utilisedare already near optimal solutions for distributed adaptivebeamforming in wireless sensor networks.

Fig. 8. Comparison of the performance achieved for a uniform distributedphase modification process and a nearest neighbour search approach

Fig. 9. Two USRP software radios constitute our current small scale scenario

IV. NEAR REALISTIC INSTRUMENTATION

We are currently working on the implemention of the sce-nario of distributed adaptive beamforming in wireless sensornetworks in a laboratory environment with USRP softwareradios (http://www.ettus.com). The optimisation logic is imple-mented within the GNU radio framework (http://gnuradio.org).The current environment with one receiver node and twotransmit nodes is depicted in figure 9. In the figure, bothtransmitters are realised by the USRP located on the left.As each USRP supports two full duplex transceivers, a singleUSRP can model two sensor nodes. We equipped the USRPmotherboards with RFX900 transceiver daughterboards thatoperate in the 900 MHz band. Channel quality measurementsare calculated by the RSSI at the receiver node. In contrast toa similar study presented in [20] the channel quality feedbackis also propagated over the wireless channel. In our currentimplementation we utilise a slightly shifted carrier frequencyfor the feedback channel to enable parallel transmission andreception by the USRP modules. With two USRP softwareradios we achieved an improvement of about 2.6dB in thesignal strength when signal phases are aligned with each other.

V. CONCLUSION

We have studied the impact of normal and uniform dis-tributed phase mutation probabilities in a closed loop synchro-nisation approach for distributed adaptive transmit beamform-ing in wireless sensor networks. In mathematical simulationsquantitative results for a network of 100 low power nodesand one remote receiver are derived. Sufficient synchronisationwas possible for both approaches within several milliseconds.We observed that minimum mutation probabilities togetherwith a suitable variance are well suited to achieve optimumsynchronisation performance.

ACKNOWLEDGEMENT

The authors would like to acknowledge partial fundingby the European Commission for the ICT project CHO-SeN ”Cooperative Hybrid Objects Sensor Networks” (Projectnumber 224327, FP7-ICT-2007-2) within the 7th FrameworkProgramme.

We would further like to acknowledge partial funding bythe ’Deutsche Forschungsgemeinschaft’ (DFG) for the project”Emergent radio” as part of the priority program 1183 ”Or-ganic Computing”.

REFERENCES

[1] G. Kramer, M. Gastpar, and P. Gupta, “Cooperative strategies and ca-pacity theorems for relay networks,” IEEE Transactions on InformationTheory, vol. 51, no. 9, pp. 3037–3063, September 2005.

[2] A. Scaglione and Y.-W. Hong, “Cooperative models for synchroniza-tion, scheduling and transmission in large scale sensor networks: Anoverview,” in 1st IEEE International Workshop on Computational Ad-vances in Multi-Sensor Adaptive Processing, December 2005, pp. 60–63.

[3] P. Gupta and R. P. Kumar, “The capacity of wireless networks,” IEEETransactions on Information Theory, vol. 46, no. 2, pp. 388–404, March2000.

[4] P. Mitran, H. Ochiai, and V. Tarokh, “Space-time diversity enhancementsusing collaborative communications,” IEEE Transactions on InformationTheory, vol. 51, no. 6, pp. 2041–2057, June 2005.

[5] O. Simeone and U. Spagnolini, “Capacity region of wireless ad hocnetworks using opportunistic collaborative communications,” in Pro-ceedings of the International Conference on Communications (ICC),May 2006.

[6] A. Krohn, M. Beigl, C. Decker, and D. G. Varona, “Increasing connec-tivity in wireless sensor network using cooperative transmission,” in 3rdInternational Conference on Networked Sensing Systems (INSS), May2006.

[7] A. Krohn, “Optimal non-coherent m-ary energy shift keying for co-operative transmission in sensor networks,” in 31st IEEE InternationalConference on Acoustics, Speech, and Signal Processing (ICASSP), May2006.

[8] C. Shuguang and A. Goldsmith, “Energy-efficiency of mimo and coop-erative mimo techniques in sensor networks,” IEEE Journal on SelectedAreas in Communications, vol. 22, no. 6, pp. 1089–1098, August 2004.

[9] L. Pillutla and V. Krishnamurthy, “Joint rate and cluster optimisationin cooperative mimo sensor networks,” in Proceedings of the 6th IEEEWorkshop on signal Processing Advances in Wireless Communications,March 2005, pp. 265–269.

[10] M. Gastpar and M. Vetterli, “On the capacity of wireless networks: therelay case,” in Proceedings of the IEEE Infocom, June 2002, pp. 1577–1586.

[11] H. Ochiai, P. Mitran, H. V. Poor, and V. Tarokh, “Collaborativebeamforming for distributed wireless ad hoc sensor networks,” IEEETransactions on Signal Processing, vol. 53, no. 11, pp. 4110 – 4124,November 2005.

[12] W. Chen, Y. Yuan, C. Xu, K. Liu, and Z. Yang, “Virtual mimo protocolbased on clustering for wireless sensor networks,” in Proceedings of the10th IEEE Symposium on Computers and Commmunications, 2005.

[13] M. Youssef, A. Yousif, N. El-Sheimy, and A. Noureldin, “A novel earth-quake warning system based on virtual mimo wireless sensor netwroks,”in Canadian conference on electrical and computer engineering, April2007, pp. 932–935.

[14] A. del Coso, S. Savazzi, U. Spagnolini, and C. Ibars, “Virtual mimochannels in cooperative multi-hop wireless sensor networks,” in 40thannual conference on information sciences and systems, March 2006,pp. 75–80.

[15] S. K. Jayaweera, “Energy efficient virtual mimo based cooperativecommunications for wireless sensor networks,” IEEE Transactions onWireless communications, vol. 5, no. 5, pp. 984–989, May 2006.

[16] ——, “Energy analysis of mimo techniques in wireless sensor networks,”in 38th conference on information sciences and systems, March 2004.

[17] Y. Tu and G. Pottie, “Coherent cooperative transmission from multipleadjacent antennas to a distant stationary antenna through awgn chan-nels,” in Proceedings of the IEEE Vehicular Technology Conference,2002, pp. 130–134.

[18] R. Mudumbai, J. Hespanha, U. Madhow, and G. Barriac, “Scalablefeedback control for distributed beamforming in sensor networks,”in Proceedings of the IEEE International Symposium on InformationTheory, 2005, pp. 137–141.

[19] ——, “Distributed transmit beamforming using feedback control,” IEEETransactions on Information Theory, (In review).

[20] M. Seo, M. Rodwell, and U. Madhow, “A feedback-based distributedphased array technique and its application to 60-ghz wireless sensornetwork,” in IEEE MTT-S International Microwave Symposium Digest,2008, pp. 683–686.

[21] S. Sigg and M. Beigl, “Collaborative transmission in wsns by a (1+1)-ea,” in Proceedings of the 8th International Workshop on Applicationsand Services in Wireless Networks (ASWN’08), 2008, (to appear).

[22] ——, “Randomised collaborative transmission of smart objects,” in 2ndInternational Workshop on Design and Integration principles for smartobjects (DIPSO2008) in conjunction with Ubicomp 2008, September2008.

[23] 3GPP, “3rd generation partnership project; technical specification groupradio access networks; 3g home nodeb study item technical report(release 8),” Tech. Rep. 3GPP TR 25.820 V8.0.0 (2008-03), 2008 March.

[24] R. Mudumbai, B. Wild, U. Madhow, and K. Ramchandran, “Distributedbeamforming using 1 bit feedback: from concept to realization,” inProceedings of the 44th Allerton conference on communication, controland computation, 2006, pp. 1020–1027.

[25] R. Mudumbai, G. Barriac, and U. Madhow, “On the feasibility ofdistributed beamforming in wireless networks,” IEEE Transactions onWireless communications, vol. 6, pp. 1754–1763, 2007.

[26] S. Sigg and M. Beigl, “Asymptotic bounds for distributed adaptivebeamforming in wireless sensor networks,” Transactions on wirelesscommunications, 2009 (submitted).